15.5.11 problem 11

Internal problem ID [2947]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 9, page 38
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:00:42 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y&=x \left (x^{2} y-1\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 45

ode:=y(x) = x*(x^2*y(x)-1)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x +\sqrt {x^{2}-c_1}}{c_1 x} \\ y &= \frac {x -\sqrt {x^{2}-c_1}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.558 (sec). Leaf size: 77

ode=y[x]==x*(x^2*y[x]-1)*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{x^2+\sqrt {-\frac {1}{x^3}} x^2 \sqrt {-x \left (x^2+c_1\right )}} \\ y(x)\to \frac {x}{x^3+\frac {\sqrt {-x \left (x^2+c_1\right )}}{\sqrt {-\frac {1}{x^3}}}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.251 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**2*y(x) - 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1}}{2} - \frac {\sqrt {C_{1} \left (C_{1} x^{2} - 2\right )}}{2 x}, \ y{\left (x \right )} = \frac {C_{1}}{2} + \frac {\sqrt {C_{1} \left (C_{1} x^{2} - 2\right )}}{2 x}\right ] \]